An operational test for the existence of a consistent increasing quasi-concave value function

The existence of an increasing quasi-concave value function consistent with given preference information is an important issue in various fields including Economics, Multiple Criteria Decision Making, and Applied Mathematics. A quasi-concave value function is equivalent to convex-to-the-origin indifference contours, a very common assumption in Economics. In this paper, we establish necessary and sufficient conditions for the existence of a value function satisfying aforementioned properties. This leads to an operational, tractable and easy to use test for checking the existence of such a value function. In addition to developing the existence test, we construct consistent linear and non-linear desirable value functions.